3.22.69 \(\int \frac {228-354 x+73 x^2+266 x^3-378 x^4+172 x^5+26 x^6-8 x^7-x^8}{-224+388 x-87 x^2-70 x^3+42 x^4-84 x^5+22 x^6+12 x^7+x^8} \, dx\)

Optimal. Leaf size=34 \[ -\frac {1}{4}-x+\log \left (5-x^4+\frac {-\frac {3}{1-x}+2 x}{7+x}\right ) \]

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Rubi [A]  time = 0.26, antiderivative size = 42, normalized size of antiderivative = 1.24, number of steps used = 3, number of rules used = 2, integrand size = 81, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2074, 1587} \begin {gather*} \log \left (x^6+6 x^5-7 x^4-7 x^2-28 x+32\right )-x-\log (1-x)-\log (x+7) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(228 - 354*x + 73*x^2 + 266*x^3 - 378*x^4 + 172*x^5 + 26*x^6 - 8*x^7 - x^8)/(-224 + 388*x - 87*x^2 - 70*x^
3 + 42*x^4 - 84*x^5 + 22*x^6 + 12*x^7 + x^8),x]

[Out]

-x - Log[1 - x] - Log[7 + x] + Log[32 - 28*x - 7*x^2 - 7*x^4 + 6*x^5 + x^6]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+\frac {1}{-7-x}+\frac {1}{1-x}+\frac {2 \left (-14-7 x-14 x^3+15 x^4+3 x^5\right )}{32-28 x-7 x^2-7 x^4+6 x^5+x^6}\right ) \, dx\\ &=-x-\log (1-x)-\log (7+x)+2 \int \frac {-14-7 x-14 x^3+15 x^4+3 x^5}{32-28 x-7 x^2-7 x^4+6 x^5+x^6} \, dx\\ &=-x-\log (1-x)-\log (7+x)+\log \left (32-28 x-7 x^2-7 x^4+6 x^5+x^6\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 41, normalized size = 1.21 \begin {gather*} -x-\log \left (7-6 x-x^2\right )+\log \left (32-28 x-7 x^2-7 x^4+6 x^5+x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(228 - 354*x + 73*x^2 + 266*x^3 - 378*x^4 + 172*x^5 + 26*x^6 - 8*x^7 - x^8)/(-224 + 388*x - 87*x^2 -
70*x^3 + 42*x^4 - 84*x^5 + 22*x^6 + 12*x^7 + x^8),x]

[Out]

-x - Log[7 - 6*x - x^2] + Log[32 - 28*x - 7*x^2 - 7*x^4 + 6*x^5 + x^6]

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fricas [A]  time = 0.86, size = 39, normalized size = 1.15 \begin {gather*} -x + \log \left (x^{6} + 6 \, x^{5} - 7 \, x^{4} - 7 \, x^{2} - 28 \, x + 32\right ) - \log \left (x^{2} + 6 \, x - 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^8-8*x^7+26*x^6+172*x^5-378*x^4+266*x^3+73*x^2-354*x+228)/(x^8+12*x^7+22*x^6-84*x^5+42*x^4-70*x^3
-87*x^2+388*x-224),x, algorithm="fricas")

[Out]

-x + log(x^6 + 6*x^5 - 7*x^4 - 7*x^2 - 28*x + 32) - log(x^2 + 6*x - 7)

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giac [A]  time = 0.21, size = 43, normalized size = 1.26 \begin {gather*} -x + \log \left ({\left | x^{6} + 6 \, x^{5} - 7 \, x^{4} - 7 \, x^{2} - 28 \, x + 32 \right |}\right ) - \log \left ({\left | x + 7 \right |}\right ) - \log \left ({\left | x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^8-8*x^7+26*x^6+172*x^5-378*x^4+266*x^3+73*x^2-354*x+228)/(x^8+12*x^7+22*x^6-84*x^5+42*x^4-70*x^3
-87*x^2+388*x-224),x, algorithm="giac")

[Out]

-x + log(abs(x^6 + 6*x^5 - 7*x^4 - 7*x^2 - 28*x + 32)) - log(abs(x + 7)) - log(abs(x - 1))

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maple [A]  time = 0.05, size = 40, normalized size = 1.18

method	result	size
risch	\(-x -\ln \left (x^{2}+6 x -7\right )+\ln \left (x^{6}+6 x^{5}-7 x^{4}-7 x^{2}-28 x +32\right )\)	\(40\)
default	\(-x +\ln \left (x^{6}+6 x^{5}-7 x^{4}-7 x^{2}-28 x +32\right )-\ln \left (x +7\right )-\ln \left (x -1\right )\)	\(41\)
norman	\(-x +\ln \left (x^{6}+6 x^{5}-7 x^{4}-7 x^{2}-28 x +32\right )-\ln \left (x +7\right )-\ln \left (x -1\right )\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^8-8*x^7+26*x^6+172*x^5-378*x^4+266*x^3+73*x^2-354*x+228)/(x^8+12*x^7+22*x^6-84*x^5+42*x^4-70*x^3-87*x^
2+388*x-224),x,method=_RETURNVERBOSE)

[Out]

-x-ln(x^2+6*x-7)+ln(x^6+6*x^5-7*x^4-7*x^2-28*x+32)

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maxima [A]  time = 0.43, size = 40, normalized size = 1.18 \begin {gather*} -x + \log \left (x^{6} + 6 \, x^{5} - 7 \, x^{4} - 7 \, x^{2} - 28 \, x + 32\right ) - \log \left (x + 7\right ) - \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^8-8*x^7+26*x^6+172*x^5-378*x^4+266*x^3+73*x^2-354*x+228)/(x^8+12*x^7+22*x^6-84*x^5+42*x^4-70*x^3
-87*x^2+388*x-224),x, algorithm="maxima")

[Out]

-x + log(x^6 + 6*x^5 - 7*x^4 - 7*x^2 - 28*x + 32) - log(x + 7) - log(x - 1)

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mupad [B]  time = 0.17, size = 39, normalized size = 1.15 \begin {gather*} \ln \left (x^6+6\,x^5-7\,x^4-7\,x^2-28\,x+32\right )-x-\ln \left (x^2+6\,x-7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((73*x^2 - 354*x + 266*x^3 - 378*x^4 + 172*x^5 + 26*x^6 - 8*x^7 - x^8 + 228)/(388*x - 87*x^2 - 70*x^3 + 42*
x^4 - 84*x^5 + 22*x^6 + 12*x^7 + x^8 - 224),x)

[Out]

log(6*x^5 - 7*x^2 - 7*x^4 - 28*x + x^6 + 32) - x - log(6*x + x^2 - 7)

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sympy [A]  time = 0.16, size = 36, normalized size = 1.06 \begin {gather*} - x - \log {\left (x^{2} + 6 x - 7 \right )} + \log {\left (x^{6} + 6 x^{5} - 7 x^{4} - 7 x^{2} - 28 x + 32 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**8-8*x**7+26*x**6+172*x**5-378*x**4+266*x**3+73*x**2-354*x+228)/(x**8+12*x**7+22*x**6-84*x**5+42
*x**4-70*x**3-87*x**2+388*x-224),x)

[Out]

-x - log(x**2 + 6*x - 7) + log(x**6 + 6*x**5 - 7*x**4 - 7*x**2 - 28*x + 32)

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